Theoretical and Experimental Constraints in the $\mu$--$\tau$ Four-Lepton Sector of the SMEFT: implications to neutrino self interactions

Abstract

We study the dimension-six SMEFT four-lepton operators in the $\mu$--$\tau$ sector. These operators control both charged-lepton scattering and neutrino self-interactions, the latter being weakly constrained by direct laboratory probes despite their importance for cosmological tensions. We compare three classes of constraints on the Warsaw-basis coefficients $[C_{\ell \ell }{]}_{2222}$, $[C_{\ell \ell }{]}_{2233}$, and $[C_{\ell \ell }{]}_{2332}$. We use perturbative unitarity from $2\to 2$ partial-wave analysis, spin-summing positivity sum rules, and the experimental bounds from NA64$\mu$ and the global fit~\cite{Falkowski:2017pss}. We find that the global fit dominates for $[C_{\ell \ell }{]}_{2222}$ and $[C_{\ell \ell }{]}_{2332}$, while NA64$\mu$ provides the leading bound on $[C_{\ell \ell }{]}_{2233}$, with the unitarity line for this direction entering the range of collider energies near $200\mathrm{GeV}$. Renormalization-group running between $1\mathrm{GeV}$ and $30\mathrm{TeV}$ modifies these coefficients by up to $10\%$. Translating these bounds onto the effective four-neutrino coupling $G_{\mathrm{eff}}$, we find values many orders of magnitude smaller than the strongly interacting regime motivated by the Hubble tension; this excludes heavy-mediator UV completions of strong ${\nu }_{\mu }$--${\nu }_{\mu }$ and ${\nu }_{\mu }$--${\nu }_{\tau }$ self-interactions within the validity of the dimension-six SMEFT and in the absence of tuned cancellations between operators, while leaving the cosmologically motivated light-mediator scenarios unconstrained by this analysis. Finally, we comment on the bounds these coefficients place on a leptophilic $L_{\mu }-L_{\tau }$ $Z^{\prime }$ UV completion. Our SMEFT-based current and projected NA64$\mu$ bounds reproduce the dedicated $Z^{\prime }$ analyses already available in the literature.

Comments: 10 pages, RevTex, 5 figures